Conditional Moment Generating Function of a Negative Binomial
$begingroup$
Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i sim NB(r-k_i, q)$, where $r$ is a constant and $K sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I have done the following:
$$M_x(s) = E[s^{sX}] = E[E[e^{sX}|K]] = E[(frac {q}{1-(1-q)e^s} )^{r-K}]$$
Since K itself is Binomial, the algebra in calculating the above expected value is complex. Is there any identity or any algebra trick that I can use to get a decent expression for MGF of X?
conditional-expectation binomial-distribution negative-binomial
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
$endgroup$
add a comment |
$begingroup$
Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i sim NB(r-k_i, q)$, where $r$ is a constant and $K sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I have done the following:
$$M_x(s) = E[s^{sX}] = E[E[e^{sX}|K]] = E[(frac {q}{1-(1-q)e^s} )^{r-K}]$$
Since K itself is Binomial, the algebra in calculating the above expected value is complex. Is there any identity or any algebra trick that I can use to get a decent expression for MGF of X?
conditional-expectation binomial-distribution negative-binomial
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
$endgroup$
add a comment |
$begingroup$
Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i sim NB(r-k_i, q)$, where $r$ is a constant and $K sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I have done the following:
$$M_x(s) = E[s^{sX}] = E[E[e^{sX}|K]] = E[(frac {q}{1-(1-q)e^s} )^{r-K}]$$
Since K itself is Binomial, the algebra in calculating the above expected value is complex. Is there any identity or any algebra trick that I can use to get a decent expression for MGF of X?
conditional-expectation binomial-distribution negative-binomial
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
$endgroup$
Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i sim NB(r-k_i, q)$, where $r$ is a constant and $K sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I have done the following:
$$M_x(s) = E[s^{sX}] = E[E[e^{sX}|K]] = E[(frac {q}{1-(1-q)e^s} )^{r-K}]$$
Since K itself is Binomial, the algebra in calculating the above expected value is complex. Is there any identity or any algebra trick that I can use to get a decent expression for MGF of X?
conditional-expectation binomial-distribution negative-binomial
conditional-expectation binomial-distribution negative-binomial
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
asked Dec 5 '18 at 2:44
Resting PlatypusResting Platypus
12
12
add a comment |
add a comment |
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